
where \(\hat{\mu }(\tau ^2)\)  is the estimated summary effect and \(\hat{\sigma }\)  the estimated standard deviation of \(\hat{\mu }\)  [1]}. These formulas illustrate the pull of the \(i\) th study effect estimate toward the estimated population mean. Hence the term shrinkage estimate. Although the posterior distribution of \(\theta _i\)  conditional on \(\tau ^2\)  is normal, in practice the amount of heterogeneity \(\tau ^2\)  is unknown and must be estimated. The resulting posterior distribution for \(\theta _i\)  will be a mixture distribution [2]}, which we denote as \(F_i\)  for study \(i\) . Two possibilities for constructing confidence intervals (CIs) are (a) define a \((1-\alpha )\) -CI via \([c_{i,\alpha /2},c_{i,1-\alpha /2}]\) , where \(c_{i,\alpha /2}\)  is the \(\alpha /2\)  quantile of \(F_i\)  or (b) numerically determine a shortest interval, which for unimodal posteriors is equal to the highest posterior density region [3]}. We apply approach (b) using the R package bayesmeta, in order to modify results from a recent study on acute kidney injury in COVID-19 patients by [4]}.
